WORKING PAPER
GRAPHICAL CONVERGENCE OF SET-VALUED MAPS
Jean-Pierre Aubin
September 1987 WP-87-083
International Institute for Applied Systems Analysis NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
GRAPHICAL CONVERGENCE OF SET-VALUED MAPS
Jean-Pierre Aubin*
September 1987 WP-87-083
* CEREYADE, Universitg de Paris-Dauphine, Paris, France
Working papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A- 236 1 Laxenburg, Austria FOREWORD
This is an introduction to graphical convergence of set,-valued maps and the epigraphical convergence of extended real-valued functions1. It is now well established that maps, and more generally, set-valued maps, should be regarded not only as maps from one space to another, but should be characterized in an intrinsic and symmetric way by their graphs. When dealing with limits of maps, either single-dued or set-valued, it is quite advantageous to wercome the natural reluctance to handle conver- gence of subsets and to replace pointwise convergence by "graphical con- vergence": Instead of studying (more or less uniform) limits of the images, one consider the limits of their graphs. One of the main reasons is that doing so is that a map and its inverse are treated on the same footing . This is quite important in approximation theory and numerical analysis. The concepts of graphical convergence of set-valued maps are related to the concepts of epigraphical limits of functions, which had recently met an important success to wercome the failure of pointwise convergence in many problems of calculus of variations, optimization, stochastic programming, etc. Finally, this report provides a first study of the Kuratowslii upper and lower limits of tangent cones, which is needed to compute general- ized derivatives and epi-derivatives of graphical and epigraphical limits of maps and functions.
Alexander B. Kurzhanslri Chairman S?rstem and Decision Sciences Program
- 'in the development of which IIASA played an important role. ... lll Introduction
Here are some basic notes on graphical limits of single-tdued and/or set.-valued maps. To deal with limits is the basis of analysis, or approximation theory, where we have to approximate objects by richer and more familiar ones. And numerical analysis is just taking up this issue for very practical pur- poses. In particular, solving equations and approximating their solutions is the ultimate task of many mathematicians. It can be regarded in this way: We want to approximate a solution xo to an equation f (xo)= yo. by approximating both the data yo by a sequence of approximatte data ynas, and the map f by a sequence of maps fn. Knowing how to find solutions to the approximate problems fn(x4)= y,, the problem is how to derive the convergence of the approximate solu- tions from the convergence of the data y, and fn and thus, to pay some attention to the limits of maps. It is nowT well established that maps, and more generally, set-valued maps, should be regarded not only as maps from one space to another, but should characterized in an intrinsic and symmetric way by their graphs. This point of view, which goes back to the protohiston of analysis with Fermat and Descartes dealing with curves rather than functions, has been let aside since for a long time. In particular. as far as limits of functions and maps go, generations of mathematicians have be accustomed to deal with many concepts of convergence of functions, from pointwise to uniform, but all based on the fact that a map is a map, and not a graph. When dealing nit11 limits of maps, either single-valued or set-valued. it is quit,e advantageous to overcome the natural reluctance to handle conver- gence of subsets and to replace pointwise convergence by "graphical con- vergence': Instead of studying (more or less uniform) limits of the images, we consider the limits of their graphs2. One of the main reasons is that doing so is that we treat on the same footing a map and its inverse. This is quite important in approximation
h his point of view of regarding maps as graphs has already allowed tn build a successful differential calculus of set-valued maps based on 'graphical derivatives'rather than set- valued limits of differential quotients. theorya, where the problem is to derive poinimise convergence of the in- verses from the point,mise convergence of the maps. Hence. a first problem is t,o study what are the relations between graph- ical and pointwise convergence. The main tool for that is the Stabiliq- Theorem4. an outgrowth of the Inverse Function Theorem for set-dued maps6. Since graphical limits are limit of graphs, which are subsets. we have to rely on the now classical concepts of Kuratowski upper and lower limits. Hence we begin by a short exposition of these concepts. With the Stability Theorem on one hand. and the concept of proper maps on the other, we are also able to complement the calculus of Kuratowski upper and lower limits of sets. and in particular. to protide crit,eria for having the natural formulas for direct and inverse images of Kuratowski limits. After defining and providing the basic properties of graphical conver- gence, we expose its applications to viability theory, where we show, for instance that the Kuratowski upper limit of viability domains is a viability domain of graphical upper limits. Finally, we relate the concepts of graphical convergence of set-valued maps to the concepts of epigraphical limits of functions, which has re- cently met an important success to overcome the failure of pointwise conver- gence in many problems of calculus of variations, optimization, sochastic programming6, etc. The use of this concept is mandatory whenever the order relation of the real line comes into play, as in optimization or Lyapunm- stability for instance. In such cases, a real-valued function is replaced by the set-valued map obtained by adding to it the positive cone (for minimization). whose graph is thus the epigraph of the function. Therefore. the convergence of the graphs of such set-valued maps is the convergence of the epigraphs of the associated functions. We present just a selection of issues dealing with epi-convergence, among
3see 112, Stability Theorem 1.11, which adapt to the general case of solving inclusions the principle stating that stability, convergence of the data and "consistency' imply the rnn- vergence of the solutions. Consistency is nothing other than graphical lower convergence. 4see [12, Proposition 1.11 &see[lo, Chapter 71 and jll] 6see for instance the book [S] and the bibliography of this took. which some formulas dealing with the epi-limits of sum and products of functions. Finally, we relate these concepts of Kuratowski limits with the ones of tangent conesi. which lay the foundations of the differential calculus of set- valued maps, and which play such an important role in optimization and viability theory. The problem we begin to consider is to stady the Iiuratowski upper and lower limits of the tangent cones to subsets K, in terms of Kuratowski upper and lower limits of sets of the form (K, - x,)/hn when h;, 3 x, - r and h, - O+. which we call respectively asymptotic paratingent and circatangent cone. One would like to relate them to the tangent cones t,o the Kuratowski limits of a sequence of subsets fin, but this seems rather difficult outside the convex realm. Let us just point out that Clarke tangent cones and Bouligand's paratingent cones to a subset K can be regarded as asymptotic circatangent and paratingent cones for constant sequences.
'B~the way. the various definitions of tangent cones are Kuratowski upper and lower limits of the sets (K - x)ih when 11 + o+ Contents
1 Kuratowski Upper and Lower limits 5
2 Kuratowski Limits in Lebesgue Spaces 10
3 Stability Theorem 11
4 Kuratowski Limits of Inverse Images 15
5 Kuratowski Limits of Direct Images 19
6 Graphical Limits 20
7 Stability of Viability Domains and Solution Maps 24
8 Epigraphical Limits 27
9 Epigraphical limits of Sums of Functions 36
10 At t ouch's Theorem 39
11 Asymptotic Paratingent and Circat angent Cones 42
12 Asymptotic Paratingent and Circat angent Epiderivat ives 48 1 Kuratowski Upper and Lower limits
We can also characterize closed set?\dued maps and lower semicontinuous set-valued maps through adequate concepts of limits. For that purpose. we introduce the following notations: we associate with a set-valued map F : S .u E' and x E ,Y the subsets I i) lims~p,~,F(f) := {y E Y I liminf$,, d(y, F(xl)) = 0} ii ) lim inf,,, F(xl) 1 := {y E E' ( lim$,, d(y, F(xl))= 0} They are otniousl?; closed. We also see at once that
(2) lim inf F (z') c F(z) c lim sup F (XI) .f +.T 9-3 The other advantage of introducing these notions is that we can define kinds of semicontinuity for "discrete" set-valued maps, i.e., "semin limits of sequences of subsets h;, of a metric space X: we set
When K, is a sequence of subsets of a metric space S,we shall also use the following notations:
\ i) Kb= lim sup,,, K, 1 ii) K\= liminf,,,, K,, Definition 1.1 When F : .X - Er ia a aet-valued map, we cay/ that lirn sup F(xl) 3'-r ia the Kuratowski (or Kuratowski-Painlevh ) upper limit of F(xl) when x1 -- x and that lim inf F (XI) 3'-3 is the Kuratonrslii for Kuratonrslii-Painle~ri) lower limit of F(xl) when x1 - x. When I<,,is a Requence of Rubsets of a metric space X, we say that
lirn sup K,, and lirn inf h', ,I'D2 n-rn are the upper and lower Kuratowski limits of the Pequence If, respec- tively. A eubset K is said to be the Kuratowski limit of the eequence K,, if K = liminf K,, = limsuph', =: lirn K,, n+oo n-w n-a; We observe at once that the Kuratowski upper- limits and Kuratondii lower limits of either F(x) or h;, and of either F(x) or h;, do coincide, since d(y:h;,) = d(y,h',). Any decreasing sequence of subsets K,, has a limit, which is the int'er- section of their closure: if Kn c Ii, when n,2 rn, then lirn K,, = n n-c<. n Remark The use of the concept of filter would avoid to duplicate these defini- tions in the discrete and continuous cases. We have preferred this longer, may be more pedagogical, solution. It is eaq- to observe that: Proposition 1.1 A point (x, y) belongs to the closure of the graph of a set-valued map F : X .u Y if and only if y E limsup9,,F(x') and F is lower semicontinuous at x if and only if F (x) c lirn inf~,, F (a') If Kn ie a eequence of eubeete of a metric epee X7then liminf,,, If,, is the set of limits of sequences a, E K,, and lirn sup,,, K,, is the of cluster points of sequences x, E h:, , i.e., of limits of subsequences X,I E Knl . It is also the eubset of cluster points of "approzimate" eequences satisfying: Then we can measure the lack of closedness (of the graph) or the lack of lower semicontinuity by the discrepancy bet?ween the values at a- of the set-valued maps F(x). liminf$,, F(xl)and lim supg,, F(xl). -4nother useful and easy consequence of the Iiuratom~skilimits is the following diagonalization lemma. Lemma 1.1 Let us consider a "double" eequence of elements x,., of a metric epace X, euch that lim ,,,, lim ,,,, x,,,, does exist. Then there ezist eequence8 n + m(n) and m + n(m) euch that Proof We set. x := lim,,,, limn,, x,., 1. Let us set x,, := limn,, x,,, and K, := {x,,,)~~~.Therefore x,, belongs to liminf,,, K, for d na. This implies that the limit x of the elements x, belongs also to liminf,,, K,, and therefore, is the limit of a sequence of elements y, belonging to K,. Such elements can be written y, = x,(,),,. 2. Let us set L, := {z,,n)n20. We observe that each x, belongs to x. Hence the limit z of the sequence x, belongs to liminf,,, L, which is equal to liminf,,, L,. Consequently, x is the limit of elements t, E L, which can be written z, = z,,(,). We observe also the quite impressive following equalities: = n(>o nq>o U$c~(t.q)B(F(y. €1 ii) lim sup,,, h;, = nnr>o U,12~rKn = n,>on~>~U,,~nrB(K~,c) iii) lim inf2,z F($) = n,,, u,,, n,,~,,,,) B(F(X'), 6) itl) liminf.,, K, = n,>Ouhr,0nn2hrB(h;l,6) By replacing the balls of a metric space by neighborhoods, we can extend through these formulas the concepts of Kuratonpski upper and lower limits to subsets of a topological space. hdam properties of closed and/or lower semicontinuous set-valued maps can be extended to the Kuratowski's limits. For instance, Theorem 1.1 Let K C S satisfy the following property: (7) for all neighborhood U of K, 3N ( Vn > ,i7 K, c U Then (8) The converse statement is true for any neighborhood U whose complement is compact. In particular, if the space X is compact , then the upper limit Kt enjoys the above property (and thus, is the smallest closed subset satisfying it). Proof The first statement is obvious. For proving the second, let y belong to the complement M of U, which is compact by assumption in the first case or because it is contained in the compact set X in the second case. Then there exist c, > 0 and Ny such that, for all n 2 N,, y does not belong to B(1{,?2cy ). Since M is compact, it can be covered by p balls B(y;, cYi). Our claim holds true for all n larger than N := mm=1,,., A\i. We also remark the following obvious properties: Proposition 1.2 Let K,,. L, be sequences of subsets of a metric space X. Then ;) lim LJ, ) c limsupn,,Kn nlimsup,,,L,, ii) liminf,,,x(K,, nL,,] c liminf,,,h;, nliminf,,,L, iii) lirn sup,,, (K, U L, ) = lirn sup,,, h;, U lirn sup,,, L,, it?) liminf,,,(h;, U L,) > liminf,,, h;, U liminf,,,, L,, t') limsup,,,n~=,h':, c nr=, lirn sup,,, Ki vi) liminf,,, nr=, KA = n~=,liminf,,,l<~ We need also to relate direct and inverse images of Kuratowski upper and lower limits to the Kuratowski upper and lower limits of their direct and inverse images. We mention now the obvious relations and postpone the proofs of criteria which transform the following inclusions to equalities. Proposition 1.3 Let K, be a sequence of rubrets of a metric space X', ,Wnbe a aequence of aubaete of a metric space Y and f : X' - 1' be a (ringle - valued) continuoue map. Then f(limsup,,,K,) c limsup,,,f(K,) I ii) limsup,,, f-'(M,) c f-'(lim~~p,,,A4,) iii) f (liminf,,, h',) C liminf,,, f (h',) it-) infl(Mn) c f-I (liminfn,,Adn) Kuratowski upper limits and Kuratowski lower limits can be exchanged by duality: Proposition 1.4 Let h', be a sequence of rubrets of a Banach epace I-. Then (11) lim,-a, inf c lim sup h',)- K, n-oi' The equality holds true when the dimension of 1Y ir finite and when the eubsets K, are cloeed conoez conee. Proof Let us choose x in liminf,,, Kt, and p in lim sup,,, h',. Then there exist a sequence of elements r, E K, converging to r and a subsequence of elements q,,~of K; converging tso p. Therefore 2 Kuratowski Limits in Lebesgue Spaces Let (fl, S,p) be a measure space and X be a finite dimensional vector-space. Let us consider a sequence of measurable set-valued maps We associate to it the subsets K, of LP(fl,X) dofined by The purpose of the next theorem is t,o compare the Iiuratowski limits of the sets K, and the sets of selections r(.) of the Kuratowski limits of the sets K,,(w). Theorem 2.1 Let us asfiume that the set-valued maps Kn are measurable and that the trubscts K, are not empty. Then {r(.)E LP(R.X) I for almost aU G. r(;) E liminf,,, K,(.c)) I c liminf,,, K,, C lirn sup ,,,, K,, I c {r(.)E Lp(R, S)I for almost all;, r(;) E lirn sup,,,, K,, (r.)) Proof I. Let x(.) belong to the first subset. Then the functions a,(.) defined b?- a,() := d(x(d).Kn (w')) are measurable and converge to 0 almost everywhere. Let us choose some y,, (.) in K,, which is not empv by assumption. Since for almost all LJ. u,(w) IJz(w) - yn(~)II and since the right-hand side of this inequality belongs to LP(!l), we deduce from Lebesgue's Theorem that the functions a, (.) do converge to 0 in Lp(!l). Let us introduce now the subsets L,(w) defined by It is clear that the set-valued map L, (.) is also measurable. The Mea- surable Selection Theorem allows us to choose a measurable selection 3, (-) of the set-valued map L, (-). It belongs to LP (!I)since for almost all w, 11zn (w)11 I IIz(WIII + an (w) Therefore z, (.) belongs to K, and converges to z(.)in LP(!l)?i.e., z(.) does belong to the Kuratowski lower limit of the subsets K,. 2. Let us choose some z(.)in the Kuratowski upper limit of the sub- sets K,,. Then there exists a subsequence of elements znt(.) of K,I converging to x(.) in LP(fl). Then a subsequence (again denoted) znt(.)converges almost everywhere to x(.) and consequently, for d- most all LJ, X(G) belongs to the Kuratowski upper limit of the subsets K,, (w). 3 Stability Theorem \Ye shall prove the following Inverse Stability Theorem which has many useful consequences. Let us recall that when x E K. we denote b?; the cone spanned b?.. K - r and by K-r TA-(r):= limsup - h-0- h the contingent cone t,o I< at x. Theorem 3.1 (Inverse Stability Theorem) Let A" and Z' be two Ba- nach spaces. We introduce a ~equenceof continuous linear operators A,, E L(X.1-)?a sequence of closed subsets h', c X. Let us consider elements 2: of the subsets h', such that both si converges to x(; and Anxn converges to yo. We posit the following stability assumption: there ezist conetant3 c > O! o. E [0, :L[ and q > 0 such that Let us set I := c/(l - cr), p < q/3E and consider elements yn and xo,, satisfying: Then, for any I' > I and n > 0, there etist solutions z';; satisfying so that ~(xo,Kt1 n ~~'(yn))5 1)1yt, - Anxon 1) (15) 1 5 11x0 - XO~I) + lll~n- YO(^ + Illyo - An~on11 converges to 0 when xon converges to xo and both Anxon and y,, E AnIC, converge to yo. Proof We choose c > 0 such that axid we consider the elements sonand yo, satisfiing (13). B?. Ekeland's l'ariational Principle (see [23] ) , we know that there exists a solution to We deduce from inequality (17)i) that SO that 11% - zoJI5 r1/3 + I(~on- zol(5 2~/3. Since y, -An 2';; E An (Ii,- 2';; ), assumption (12) implies that there exist un E TKn(G) and v, E 1'- satisijing By definition of the contingent cone, there exist elements h > 0 and eh E X converging to 0+ and 0 respectively such that By taking in inequality (17)ii) such an x,, by obse~ngthat y, -Anxn = (1 - h)(y, - A,G) + hw, - heh, we deduce that Dividing by k > 0 and letting h (and thus, eh) converge to 0, we get: Since we have chosen E such that Q + cc < 1, we infer that 2';; is a solution to 2, E h;, &Anz';, = y,, satisfying As a consequence, n7e obtain the following important statement. Theorem 3.2 (Inverse Function Theorem) Let X and Z' be two Ba- nach spaces. We introduce a sequence of continuous linear operators A,, E L(X.I-)? a eequence of cloeed eubsets Kn C X. Let us coneider elements xt of the pubeets Kn euch that both rt converge8 to x: and Anxn converges to yo. We popit the following stability assumption: there ezi~tconstants c > 0, a E [0,.1.[and q > 0 such that Then for any eequence of elements xon of hrnconvetging to xo and euch that A,(xon)convetgea to yoand any sequence of element8 y, E Y converging to yo, we have d(zon,Kn n f;'(Yn)) 5 lI(Yn - fn (~0n)11 The stability assumption (21) implies implicitly that xo belongs to the lim inf of the subsets K,. We consider now the lim inf of the contingent cones8 T (so):= liminf Txn(xn) = n U TK,,(xn j CB Kn3xn4~0 n + c-0 N,r) n>N,s,EK,n(x+qB) and we address the following question: under which conditions does the "pointwise surjectivity assumptionn imply the abcrve stability assumption of the Kn. The next result answers this question when the dimension of E' is finite, unfortunately. Proposition 3.1 (Pointwise Stability Criterion) Aeeume that T(xo) is conve28 and that AT(xo) = E'. Then there etiete a constant c > 0 euch that, for all a €]O,l[? there eziet q > 0 and -V 2 1 with the following 'which is equal to the asymptotic circatangent cone when the dimension of X is fhitt. See Proposition 11.2 below. 'This is the case when the dimension of X is bite thazrks to Proposition 11.2 below. property: V t: E I-, V n 2 ,V. V x,, E Ii,,n (ro+ qB). there ezisf eolutione u,, E TK,,(x,,) and w, E Y to Proof Let S denote the unit sphere of I-, which is compact. Hence there are p elements t7,such that the balls ti, + fBHcover S. Since T(x0)is convex and AT(xo)= 1'. Robinson-Ursescu's Theorem implies the existence of a constant X > 0 such that we can associate with any tT,E S an u, E T(xO) satisij&q Ilu,llz 5 A. By the very definition of T(xo),we can associate with o €10. I [ integers N, and q, > 0 such that V n 2 N,,V xn E li,n(xo+etaiB). there eldst u; E TKn(x,,) satisfying Let AT := max,~,~pATi and q := minrliLP q;. We take n 2 iV and x,, E K,, n (so+ q B). Let v belong to Y. There exists tT;E S such that Set vn = I~v11 y~.:and Wn = v - Awn. We see that vn E TKn(xn)? that (where e := X + ((Al(r(2,yj/2)and that This proves our claim. 4 Kuratowski Limits of Inverse Images We have seen that, inclusion liminfn-+m /-'(.$in)c f-I (lirninfhi,,)n-w is always true. We can provide sufficient conditions for having the equal- in- in the case of lower limits. For instance, the usual Liusternik's Inverse Function Theorem implies the following proposition: Proposition 4.1 Let us assume that S and 1' are Banach spaces, that the map f is continuously dinerentiable at some point r E f-l (liminfn-oo M,) and that f'(x) is surjective. Then (23) x belongs to liminfn -k-, f-'(iVn) But, before extending this result to more general situations (when ,Y is replaced by a subset K, for instance), let us proceed with the simpler case of convex subsets. Proposition 4.2 Let us coneider two Banach space8 X and 1' , a continu- ous linear operator A E l(,Y, Y)and two sequences of subsets L,, c X and Afn c 1.. We assume that 6) L, and Mn are convex (24) ii) L,, are contcu'ned in a bounded ~et iii) 37 > 0 I -jB c A(L,) -itin Then (25) limn-w inf (L, n A-' (MI,)) = limn-w inf L, n A-'(lirnn-K inf ,Idn) Proof The inclusion liminf(L, n~-'(M,)) c A-'(liminfL, nliminfiti,) n-a; ' n-oo #A-0; being obvious, let us prove the other one, by checking that any x in lirn inf L, n A- ' (lim inf iLfn ) 11 -+o: n-03 is the limit of a sequence of elements x, belonging to Ln such that .S(z,,) belong to hf,, . \Ye know that r can be approximatbed b?. elements u,, E L,, and that A(x) can be approximated by elements un E All, ,,. Then c, := ~JA(U,,)- tl,, 1) converges to 0 and 0, := & converges to 1, belongs to ]0,1[ and satisfies O,,E,, = (1- O,,)?. Therefore. and consequently, there eAst elements uk E L, and r: E M, such that If we set x, := 9,un + (1 - 9,)uI,, we observe x,, belongs to L,, and that .4(xn) belongs t,o hi, for these subsets are corrvex. Furthermore, llx, - u,, 11 = (1 - 9,)11un - uI, (1 converges to 0 since u, and uk remain in a bounded subset by assumption. Remark Assumption (24) implies obviously that For non convex subsets L, and M,, we obtain the following consequence of the Irrverse Stability Theorem 3.2: Theorem 4.1 Let X and Y be two Banach #paces. We introduce a contin- uous linear operator A E l(X,Y) and sequencee of cloecd eubects L, c X' and Adn c Y.Let ue aeeume that there eziat conetanta c > 0, o E [O,l( and q > 0 euch that Then the Kuratoweki lower limit of L, n A-' (M, ) is equal to the inter- section of the Kuratowaki lower limit of L, and the inoeree image by A of the Kuratoweki lower limit of M,: liminf(L, n A,'(M,)) = liminf L, n A-' limidilf, n-03 A-03 n Proof Since the inclusion liminf(L, n ~;'(,kf~))c liminf L, n A-'(liminfA4dn) 11 -m n+ci 11 +x is obvious, let us take any xo := lirn,,, x,, belonging to liminf ,,,, L,, such that yo := Axo = lim Axon= lim yon n+cc n--oc belongs t,o liminf,,, M, . We then apply Theorem 3.2 to the subsets L, x itif, of S x 1' and the continuous linear operators A 6 1 associating to any (x,y) the element Ar - y, since we can write The pair (so,. yotl ) E Ln x Mn converges to (q,yo), and (A 6 1)(xOn, yon ) converges to 0. Furthermore, it is clear that assumption (29) implies the stability as- sumption (21) of Theorem 3.2. Therefore, by Theorem 3.2, there exits a solution (z, y", ) E L, x Mn to the equation (A a l)(<, z)= 0 such that This means that % belongs to h',? converges to xo and that A& converges to Yo. Remark We can extend this theorem to the case of a sequence of continuous linear operators A,, E l(X,Y), where we take i) xo := limn,, x, E liminf.,, L, (31) \ 1 ii ) yo := lim,, ,, A, xo, = limn+, yo, E lim inf ,,, hin The same proof where A is replaced by A, implies that 3 E K, such that 1;) 6 - 2.0 1 ii) A,g - Yo 5 Kuratowski Limits of Direct Images We have seen that inclusion f (lim sup fit,) c lim sup f (It;,) n-+w tr-+w is always true. ?Ye obtain equalities for the Kuratowski upper limits when f is proper'0 Proposition 5.1 Let us assume that f is proper, then (33) f (lim SUP Kn) = lim SUP f (Kn ) n-m n-+m and if f is proper and su jective, then (34) limsupn-m f-'(Afn) = f-' (limsupA!fn)n-+m We can adapt the Closed Range Theorem" to obtain the following equality: (We denote by the polar set of K). Theorem 5.1 Let X and 1' be reflezive Banach spaces, Kt, c ,Y be a subsets and A E L (X,Y) be a continuous linear ~perator'~satisfying ''We recall that a continuous single-valued map from a metric space X to a metric space 1' is proper if and only if one of the equivalent statements If /(z,)converges in 1' then a subsequence of z, converges in S j maps closed subsets to closed subsets ii) Vy E 1'. j-'(y) is compact "see 110, Theorem 1.5.5, p261 '"anach's Closed Graph Theorem allows to assume that A is surjective: It is sufficient to decompose -4 as the produrt A 0 6 of the canonical surjectinn CJ from X nntn its factor space S/ker(,4) and the associated bijective map -4, which is an isomorphism. Then the properness of -4 is equivalent to the properness of 6. Let K: denote the Kuratowski upper limit of the subsets I{, when S is supplied with the weak topology. Then Proof Let us consider a sequence x, E Is;, such that A(x,) con- verges to some y in 1'. We shall check that this sequence is weakly bounded, and thus, weakly relatively compact. Let us take for that purpose any p E X*,I1pII. 5 7,which can be written. by assumption ( 35) Therefore, there exists a X such that T E n,,,N IC," and consequently, since the converging sequence Ax,, is bounded. Then a subsequence (again denoted) converges weakly to some x which belongs to Xi. o 6 Graphical Limits We shd use these concepts to define graphical convergence of set-valued maps. Definition 6.1 (Graphical Convergence) Let up consider a sequence of set-valued maps Fn : X Y. We shall Ray that the pet-valued mapa Fc and Fyrom S to I' defined by \ i) GraphFi := limsup,,, Graphf,, (39) 1 ii) GraphFb := liminf,,, GraphF,, are the (graphical) upper and lower limits of the set-valued mapa F, reepectivelg. We praide a more explicit characterization of these graphical upper and lower limits. which follows immediately from Proposition 1.1. Proposition 6.1 Let ua consider a sequence of set-valued maps Fl, : S .L. Ir. Then y belonga to FF(x)if and only if it ia the limit of a subsequence of elemente y,~E F(x,,,) where r,,~converge8 to x. It belong8 to F~X)if and only if it is the limit of a aequenee of element8 y, E F(r,,)where r,, converges to x. Let us point out these useful formulas: Proposition 6.2 Let US consider a sequence of set-valued maps Fn : S .L. 1'. Then ,J Fb(x) c n,,, lim infn-, Fn(B (2, €1 ) F)2 n,,olimsu~n-, Fn(B(x. €1 1 These formulas can be regarded as relating graphical convergence aith some kind of " almost pointwise convergencen. But can we compare the graphical convergence of Fn and the ?'pointwiseconvergence " of Fn,i.e., the upper and lower K~rat~owslii'slimits of the subsets Fn(x)? The following statement provides the easy answers. Proposition 6.3 Let us consider a sequence of set-valued map8 F,,: X .L, Y. Then the following relations hold true: The missing equality holds true under more assumptions. Theorem 6.1 Let X and I- be two Banaeh spaces. We consider a sequence of aet-valued maps Fn : ,Y .u Y and its upper graph limit FPdefined by Let us consider yo E F"so) and let us assume that there etiet conetants c > 0, cr E [0,I[ and q > 0 such that Hence, for any sequence zn converging to xo, we have yo E liminf Fn(z,) n4x Proof \Ye apply the Inverse Stability Theorem 3.2 to the case when the subsets h;, are the graphs of the set-valued maps Fn, when iV,, are the singletons {r,) and when the continuous linear operator A is the projection R-Y from ,Y x lr onto ,Y. since F,)= r,- (~ra~h(~,)nr\.l(z,)'j The uniform boundedness of the contingent derivatives on a neighbor- hood of (so,yo) implies obviously the stability property (21) uith o = 0. E! \Ve can translate the Inverse Stability Theorem 3.2 into the following useful statement: Theorem 6.2 Let us consider a sequence of set-valued map8 Fn : X .u Z': an element (xo,yo) of the graph of ita graphical lower limit and let us aaaume that there ezist constants c > O? o. E [0, I[ and q > 0 such that v (xn: yn) E Graph(F71) n B((x07YO), 9): I Vt7 EY, 3un EX, 3 w, €1' such that v E DF,(x,,y,)(u,)+w, I and ~lunll < cll~ll ll~inII < 011t711 Then?for any sequence (so,, Yon ) converging to (~o,yo), for any Yn converg- ing to yo, we have Proof R7eapply the Inverse Function Theorem 3.2 with X replaced b?- ,Titimeal', h;,y Graph(F,,): A by the projection Ill, from X x 1' onto 1'. We have to prove that assumption (21) of Theorem 3.2is satisfied. i.e., that for all r E 1.. there exist (u,,. t~,)in the contingent cone TGraphlFnl(x,, yll ) and w, E lr such that t: = v, + wn, max(llun (1, JJvn11) < cllt-ll and IItcrI11 5 a (It.(J.These informations are provided by our assumption sinre the contin- gent cone to the graph is the graph of the contingent derivative and since ((t~n=t~-u~n((i~~maUerthenorequalto(l+a)Jlt*Il. An important consequence is the Inverse Function Theorem for nonlin- ear constrained (single-value)d maps. Theorem 6.3 (Inverse Function Theorem) Let X and 1' be two Ba- nac h apaces. We introduce a aequence of continuoua single -valued maps f,, from X to 1- a sequence of closed subsets I<, c S and an element (xo,go in the graphical lower limit of the retn'ctions of f,, to the subsets h;,. We assume that the functions fn are diferentiable on a neighborhood of xo and we posit the following stability assumption: there etist conetants c > 0, a E [O. I[ and q > 0 such that Then for any sequence of elements xo,, of K, converging to xo such that fn(~o,,)converges to yo and any sequence of element3 y, E Y converging to yo, we have onf) I I(Jyn - fn(zon)l) Proof It is sufficient to recall that the contingent derivative of the restriction F, := fniKn of f, to K, is the restiction of the derivative f,',(x,,) to the contingent cone TK,(xn)to K, at 2,. 17 Remark Since the abwe theorem implies obviously Theorem 3.2, nre infer that all these statements are equivalent. 17 hlIonotone and Maximal Monotone Maps do enjoy interesting properties. For instance, it is sufficient to know that the graphical lower limit of a sequence of monotone maps is m&mal monotone for deducing that it is actually the graphical limit: Proposition 6.4 (Graphical Convegence of Monotone Operators) Let X be a Hilbert space. We euppose that the set-valued mape F,, : X .u S* are monotone and that F : X .u X* is mazimal monotone. If F is contained in the graphical lower limit Fb of the F,, '8, then F is actually the graphical limit of the F, '8. Proof We have to prove that the graphical upper limit Fg of the set-valued maps F, is contained in F. Let p belongs to FS(z).Hence the pair (2,p)is the limit of a subsequence of elements (x,i, p,i) of the graph of F, . Take nonr any pair ( y ,q) in the graph of F. Since F is contai~ledin the graphical Iiuratoa~skilower limit F" assumption, we know that there exists a sequence of elements (yn,qn) of the graph of F, converging to (y, q). The monoticit\- of the set-dued maps F,, implies t,he inequalities Going to the limit, we deduce that Therefore p belongs to F(x) because of the maximality of the graph of F among rn~not~onegraphs. 7 Stability of Viability Domains and Solu- tion Maps Let us consider now a sequence of closed viability domains of a set-valued map F". Does the Kuratowski upper limit (see Definition 1.1) of these closed viability domains is still a closed viability domain? The answer is positive. Theorem 7.1 (Stability of Viability Domains) Let us consider a nun- trivial upper semicontinuous set-dued map F : X - X with compact convex images and linear growth . Let K,, be a sequence of closed viability domains of F. Then the Kuratowski upper limit ia also a closed viability domain of F. 13sce 1131. A subset K c Dom(F) is a viability domain if and only if The Viability Theorem states that for upper semicontinuous set-valued map with nonempty compact convex images and with linear growth, A' is a viability domain if and only if K enjoys the viability property: Foe all initial state in K, there exists a solution I(.) to the differenrial inchlsic,n r' E F(z) which is viable in K. Proof We shall prove that ICE enjoys the viability prope:~. Tlle necessary condition of the I'iability Theorem14 implies that this subset is a viability domain. Let r belong to KY It is tlle lixuit of a subsequence sf,!E Kl,~.Since the subsets K, are ~iabiliqdomains. there exist viable solutions y,,l(.) to the differential inclusion x' E F(x) starting at xnt. The upper semicontinu- ity of the solution map implies that a subsequence (again denoted) ynl(.) converges uniformly on compact intends to a solution y (.) to differential inclusion x' E F(x) starting at x. Since y,~(t) belongs to Knr for all n'. we deduce that y(t) does belong to Kt for all t > 0. 0 Since we are dealing aith Kuratowski upper limits, the question arises whether the Kuratowski upper limit Ke of a sequence of closed viability domains I{,, of set-valued maps Fn is a closed viability domain of the closed convex hull of the upper limit mF%f the set-valued maps Fn defined b\. V x E X, (mFg)(x) := =(FG(z)) Theorem 7.2 (Stability of Solution Maps) Let us consider a sequence of nontrivial set-valued maps Fn : X .u A' satisfying: Then 1. The Kuratowski upper limit of the solution maps SF,, is contained in the solution map ScoFa of the co-upper limit of the set-valued maps F, 2. If the subsets li, c Dom(Fn) are closed viability domains of the set- valued maps Fn:then the Kuratowski upper limit Kd ia a closed via- bility domain of coFa. 3. The Kuratowtki upper limit ofthe viability kernels of the set-valued maps F,, is contained in the viability kernel ofcoFfi. It follows from the adaptation of the Convergence Theorem to limits of set-valued maps. 14see I?, Theorem 4.2.11. Theorem 7.3 (Convergence Theorem) Let S Be a topological vector space. 1' be a finite dimensional vector-space and Fn be a sequence of non- trivial set-valued map8 from X to 1'. Let us assume that the set-valued maps F,, are uniformly bounded. Let I be an interval of R and let UP consider measurable functions x,, and y, from I to S and Y respectively, eatisfying: for almost all t E I and for all neighborhood U of 0 in the product space X x l', there eziets Ad := M(t,V ) such that If we assume that i) x,(.) converges almost, everywhere to a functionx(-) ii) y, (-) E L1(I,Y; a) and converges weakly in L1(I,I'; a) to a function y E L1(I, E': a) then (46) for almost all I. E I, y(t,) E co(Fr(~(t )) Proof The proof is a straightforward extension of the Convergence Theorem and of the following Lemma: Lemma 7.1 Let ua consider a sequence of subsets Kt, contained in a bounded subset of a finite dimensional vector-space X. Then (47) m(limsupK,,) = CT( K,, j n-+m N>On n2hru Proof The closed convex hull of the Kuratowski upper limit is obviously contained in the closed convex subset We have to prove that it is equal to it when the dimension of S is finite and the subsets Kn are contained in a bounded set. Since an element x of -4 is the limit of a subsequence of convex com- binations a2y of elements of U,>N K, and since the dimension of X is an integer p, CarathGodory's Theorem allows t,o write that where S, 2 S and where. +A; belongs to Ii!,:,.. The vector a,.' of y + 1 components o? contains a converging subsequence (again denoted) a" which converges to some non negative vect.or a of p + 1 components a, such that S,P=,a, = 1. The subsets Ii, being cont,ained in a given compact subset. ure ran ex- tract successivel>. subsequences (again denoted) x.z., converging to elements x,, which belong t'o the Kuratowski upper limit of the subsets li,,. Hence r is equal t,o the convex combination ~~=oa~xjand the lemma is proved. Remark L€ we dont assume t,hat the set-t-alued ma,ps F,,are uni- formly bounded, then we cannot use the above lemma. However. we can conclude that \ for almost all t E I. - I Y(O E n.,o,~>oCOU,,~,.~, , pn 8 Epigraphical Limits Let us consider a sequence of extended real-valued functions whose domains are not empty. For taking into account the order relation of R, we associate with ea,ch extended real-valued function I.', two new set-valued maps V,, and Vnl defined in the following way: lL(x) RL if r E Dom(li) i) Vn; := 1 + 1 fl if x + Dom(l:, ) Tk(x) - RI if x E Dom(Tl,,) ii) V,, := I I 1 fl if x + Dom(1;) We see at once that Therefore. using the concept of graphical upper and lower limit with these mro associated set-valued maps. nre come up with the concepts- of epi and h?-po convergence. which are thus obtained by taking Iiuratowslii upper and lower limits of their epigraphs and h>.pographs. Definition 8.1 (Epi-limits) Let us consider a sequence of ezfended red- valued functions I,',i : S r RU {+=) whose domains are not emptg. We shall sag that 1. the function 1': whose epigraph i.r the Kuratowski lower limit of the epigraphs of the functions Ti (49) Ep(T.\') := limt,-cx inf Ep(I ,) is the upper epi-limit of the functions I;, 2. the function whose epigraph is the Kuratowski upper limit of the epigraphs of the functions Vn (50) ~~(1:) := lim supEp(1,) 11-0s is the lower epi-limit of the functions I.', 8. the function I.-1'whose hypograph is the Kuratowski lower limit of the hypographs of the functions Vn is the lower hypo-limit of the functions 1,; 4. the function 1': whose hypograph is the Kuratowski upper limit of the hypographs of the functions I';, Hp(l'l) := lim sup Hpjlh) 11 -Cc is the upper hypo-limit of the function^ I;, If the upper and lower epi-limits coincide, we shall sag that the common value (53) I; := 7; = r: is the epi-limit of the sequence of functions 1;: and we define the hypo- limit 1' in the same way. The terminolop concerning the epi-limits seems at odds with the choice of the Iiuraton4ii semi limits: the upper epi-limit is associated n-it11 the Iiuraton-slii lower limit. However. they are consistent in the case of h~po- limits. This is due to the anal:~ic.al definitions of these epi-limits. invohing the concepts of I?-convergence and lim sup id. defined in the following way: Definition 8.2 (Lim sup inf) Let L and be two metric spaces and dl : L x A4 I- R be a function. We set (34) lim sup2,, inf,l,,q(x', y') := sup inf sup inf 6(xf,y') 00 'I~O~~~B,,.~,~y'EB(y.0 The concept of lirn inf sup is defined in a qmetric vay. and the adaptation to sequences (or filters) is straightforward. Proposition 8.1 Let us consider a sequence of eztended real-valued func- tions 1.1, : ,Y I- R U {+m) whose domains are not empty. We obtain the f olloloing formulas: i) l;"(xo) = lim sup,,, inf,,, I;, (x) ii) l,-;(xo). . = lim inf, ,,.,,,o 1:, (x) ;ii) VlD(xo)= -(-V)!(x) = lim inf,,, sup,,, la (2) i.) I~;(X~)= -(-I*);(x) = lim sup,,,,,-, I,(x) Pro of We shall check these formu1a.s for epigraphical convergence onll-. I. For computing the value of T:b at xO,we use the fact that for every X 5 1.': (xO),there exist sequences of elements x,, converging to xo and An to such that A, 2 I,;, (x,,). Therefore, for all c > 0 and 7 > 0, there exists AV such that, for all n 2 N, nre have inf l;,(x) (x,)I A,, _< X+c i/-~-q, /:ct1 and thus sup inf I;;, (x) 5 X + f ,>A'- Iix-xo,iS~ from which we deduce that V X 2 17,'(so), lim sup inf I ;, (x) 5 X n+m "'0 and thus, that limsup inf 1.:,(x) I l.r;(a-~) ,+CY\ .?+.?o 2. Conversely, for pro~ ( 56) n +W,X-.TDmi I) 5 l~'(s0) 4. We conclude by observing that the pair (so,X1) where X1 := liminf I;,(x) n-w,x-+~ belongs to the epigraph of 1';. because, by the very definition of the lim inf, we can construct a subsequence of elements (again denoted) (x,, A,) of the epigraph of T', converging to (xO.XI). It ma?. be useful to store these inequalities: Remark We have defined the concepts of epi-limits from the Iiu- ratlowski limits of the epigraphs. Conversel>-. we can recover the Kuratowski limits of subsets from the epi-limits of their indicators. Proposition 8.2 Let ua consider a sequence of subaets h;, c S and their indicators ul~,. Let Khnd KVenote the Kuratowski upper and lower limits of the K,, '3. Then 1 i) VKb is the lower epi-limit of OK, (58) 1 ii) i.K~is the upper cpi-limit of iyKn Katurally. we can introduce the same definitions for "continuous" pa- rameters u E U.where U is a topological space. In such a framework, we consider a family of extended real-valued func- tion lT(u): ++RU {+w) depending upon the parameter u, t,o which associate the setvalued maps (u, x) + R+ if (x, u) E Dom(1') if r 4 DO~(T*~) V(u,x) - R+ if (x.~)E Dom(TT) if x 4 Dom(1') There is a subtle, but important, difference between the extended red- valued function V : (u,r) € U x X I- R U {+oc), for which the variables u and r are on the same footing, and the setrvalued maps V(.)t : u E U - V(u)?,for which the ~ariableplay a different role: u, the role of a parameter, whereas the order relation on R involves the variable x, when we minimize the function with respect to x, for instance. To emphasize this difference, the settdued map V(.), is called a vari- ational system. Definition 8.3 (Variational system) Let us consider a variational sys- tem V(-)tdefined on U x X. We shall say that I/'(.. .) (or V(.)t): to be preciee, is 1. upper epi-continuous at u if the set-valued map u' - V(u1)?is lower semicontinuous, i.e., if and only if Lr(u,Z) := limsup inf I*(ul,x') ,,l-u 2-3 2. lower epi-continuous at u if the graph of the eet-valued map u' - V(u1)?is closed ? i.e., if and only if IV(u,z) = liminf FT(u,rr) ul+u.Z'-3 3. epi-continuoue at u if the set-valued map u' ++ V(U')~is lower eemicontinuous and has a closed graph, i.e., if and only if (62) 1' (a,x) = lim sup inf 1' (u', x') = lim inf 'C7 (u, x) ,p+,, 9-3 ul-u.2-3 The definition for hypo-continuity and semicontinuity are naturally q7m- metric. We expect that the infimum of an epi-limit is closely related to the limit of the infima. This is detailed in the following statement: Proposition 8.3 (Limits of Infima) Let us consider a sequence of ez- tended red-valued functione V,, : ,Y R U {+oo) whoee domain8 are not empty. Then lim sup(inf V, (x)) n 3EX 5 21- Trip If we assume that the function8 I,: are lower semicontinuous and uniformly inf-compac t ? then ix-$-I8-; 5 lim inf (in[- l, (x)) 3E n-oc '3E. Coneequently, if V is the epi-limit of a sequence of lower semicontinuous uniformly inf-compact functions '5/, , then j i) infxEx Y (x) = limn+m (infZE~-1; (x)) (63) ii) lim {xtllKl (xn) = inf35.\-Vi (x)) c = infXEx17(x) Proof 1. Let A > infZEx1;b(x) be fixed and xo be chosen such that 1'; (xo)< A. We know that for all q > 0, there exists AT > 0 such that sup inf I)5 J?(z,) t,>h' zEB~zo,~j Therefore, sup inl-b, (x) < sup inf 1. (a) n>N n2N *Bl~o,t)) so that inf,,x ITn(x) 5 A. Hence it is enough to let N go to m and A to iq(xo). 2. Let, us consider a subsequence (again denoted) C', such that On the other hand, since the functions 1,; are inf-compact, the min- ima are achieved: there exist x,,'~such that inf* l/,, (x) = V,,(x,,). They remain in a relatively compact subset since the functions v, are uniformly inf-compact. So a subsequence (again denoted) xn does converge to some xo. Therefore, inf,x?(x) < b?(xo) = liminf ,+,,,,, ,K(x) < infn+mvn(xt,) = lim inf ,+, infrcx- Vn (x) 3. If we set Fn(x) := Vn7(x) ale see that the level sets of 1/;, are the inverse images Fn-'(A) of F,. Since we know that F~-'(x) = limsup F;'(X,) n--+w,X,+X we deduce that the level sets of the lower epi-limit, are the Kuratowslii upper limits of the level sets: (11 lY(x) < A} = Ern sup {z 1 1; (x) 5 n +co.X, -X By taking X := idfix-Lr(x) and A, := infxEx-1; (x),which converges t,o X by the two first stat.ements of the Proposition. we infer the third one. ITnfortunately. there are counter-examples for the property that the set of minimizers of the upper epi-limit is the Kuratowski lower limit of the sets of minimizers of the functions V,. However, the Stability Theorem praides some results about the Kuratonrski lower limits of level sets, but which exclude the case when the level sets are set of minimizers. Proposition 8.4 Let ue coneider a eeguence of eztended real-valued func- tions V,, : I- I?. U {+oc) whoee domains are not empty. Let ua aeaume that there ezist xo, X > 0 and conetante c > 0, q > 0 euch that , for all n 2 N, x E B(xO,q): i) 3 U, E cBx euch that DT(Vn)(x)(u,)= -1 (64) ii) 3uX E cBy euch that DT(V,,)(x)(uX)= +1 Then there eziet a constant 1 such that, for any eequence of element8 so, converging to xo and such that Vn(zon)converges to V;(xo),and for any An converging to (20), there ezist eolutione 6 eatiefying (65) ii) IIE - xon11 < llXn - V; (so)1 Proof We apply the Stability Theorem 6.1 to the inverses of the set-valued maps F, (x):= Vnt(x).We have to check that is bounded by c. It is enough to choose u = u,' when p = +1 and u = u; when p = -1. Hence assumption (40) of Theorem 6.1 is satisfied, and its conclusions imply the conclusions of the above theorem. Remark Observe that these stability assumptions (64)imply that for all n 2 N and all x, E B(xo,q), since the Fermat rule is violated. We can also derive from the Inverse Stability Theorem 6.1 criteria which imply some equalities in formulas (57). Proposition 8.5 Let us consider a sequence of eztended real-valued fune- tione I;, : X I- R U {+oc) whose domains are not empty. Let us assume that there ezist xo and conrtants c > 0, n' > 0 and q > O such that . for all n 2 N. x E B(xO,q),the domaine of the contingent epiden'watiwee DT(Vn)(x) are equal to the whole space and (67) sup lDt(vn)(x)(u)l 5 c llu/l=l Then (68 I;(.) = I?(-) Proof We apply Theorem 6.1 to the set-valued maps F,, defined by It is easy to check that assumption (67) implies Theorem 3.2's stability assumption (21). This is straightforward when A, = V(x,),since llDF,(x,A,)ll := sup inf 1111 5 sup lD~(I'k)(x)(a)l5 c II~I/=~p'DFn(+Xn) Ilu l/=1 When A, > V(x,),we know that Dorn(D+(V,)(x))x R c Graph(DF, (x,A,)) so that (1 DF, (2,A,)IJ := sup inf IpI = 0 11, /j=l PEDF~(J,X~) Hence, there exists a constant c such that, for all n 2 L'V , x E Dom(I<,) close to xo and A, 2 I; (x)close to T.;D(xo),we have Kow, to say that V/(so) belongs to the Kuratowski lower limit of the V,I (x) when n -) oo and x -+ xo implies that lim sup ITn (z) 5 T.~(X~) n -w ,x-x~ 9 Epigraphical limits of Sums of Functions Let us consider two Banach spaces S and Y,a continuous linear operator A E l(S,E'), amd two sequences of extended real-valued functions F.', : -Y + R U {+m} and IT; : E- + R U {+m). Re shall compute the upper epi-limit of the functions in tmermsof the upper and lower epi-limits of the functions 1 ;, and TTTn . Theorem 9.1 Let us consider two Banach spaces X and Y,a continuous linear operator -4 E l(X,Y), and two sequences of eztended real-valued functions I-',, and W, form X and 2' to RU {+oo) reepectively. 1. The following inequality hold8 true. 2. We posit the following stability assumption: there exist constants c > 0, CY E [O. I[ and q > 0 such that, for all n, ) V x E Dom(J;,) n B(x,. ol), V y E Dom(CT;,) n B(Axo,ol) I BI- c -4 (~om(Di(16)(r)) n e~-l-)- Dom(DT(m:) (y)) + oBy ) SupaEDom(~;i\;)i,lllDi('; )(r)(u)I/IIuII 5 I iii) s"ptj~D~mi~r[K-,,)(~)I n 1 (/5 c Then, the upper epi-limit U,P of the sequence of functions U,,:= 1.6 + W,,oA eatisfiee the estimate: Consequently, if the sequences of functions 1/, and CT:,, have epi-limits 11' and W.' respectively, so does the sequence of funetiona U, := vi + IT', o A and its epi- limit U satisfies Proof 1. Since inequality (69) holds true when one of the values ~~~'(x.)and IV~(AX~)is equal to -m, or U:(I~) is equal to +a.we have to check thb formula when the two first values are larger than -oc and the last one smaller than +a.Then, there are finite number p, integer N and positive number 7 such that., by definition of the li1.u inf. Vn.2 N, Vr~B(x~,q),l,',(r) and W,,(.4r) > y By definition of the lower epi-limit, the pair (zo,u!(zo)) is the limit of a subsequence (xn,c,) satisfying The t 2. We begin by obsening that if we set i) K := Ep(1') x Ep(1T') x R c X x R x E' x R x R ii) G(x, a: y, b, c) := (-42 - y, a + b - c) iik) H(x,a, y, b,c) := (x,c) we can write (71) Ep(U) = H (Kn G-'(o,o)) Therefore, it is sufficient to show that, the Kuratonrski lower limit of the subsets K,, nG-'(0,O) contains the intersection of the Kuratowslii lower limit of the subsets Kn with G-'(0,O). For that purpose, we shall use Theorem 3.2. Let us consider any sequence of elements xon, sol,, yon, and bo, con- verging respectively to so, V{(xo), .4x0 and Wi(Axo) and let us set Con := son + ban We obseme that the elements (son, son, yon , bonycon ) belong to K, and that G(son,son , yon , ban, con) = (Aton - YO,, 0) converges to ( 0, 0). We begin by checking that the assumptions of our Theorem imply the stability assumption (21) of Theorem 3.2, i.e., that there exists a constant c > 0 such that, for all n, for all for all (z,A) E X x there exist (u,p, v,v, 6) and e such that Assumptions (9.l)i)& ii) imply right away that Let us take now p := c((ull,v := cllvll and 6 := c(((u((+Ilvll) -A. We deduce from (9.l)iii) that (u,p) belongs to Ep(D;V,)(x),that (r,v) belongs to Ep(DtWn)(y) and that 161 I ()A(+ c(ll~II+ 1~1))I cf(II--ll+ /A(). The conclusion of Theorem 3.2 being available, we then know that hhhA h there exist elements (3, ,a,, yn , b, , cn ) E h;, satisfying Therefore, we infer that h since both ao, and ir';; converge to v(.co) and both botl and b, converge to W:(Axo).Such fact implies the inequality (70)we wpre looking for. 10 Attouch's Theorem Theorem 10.1 (Attouch's Theorem) Let us consider a sequence of ez- tended real-valued lower semicontinuous convez functions 1.; : .X - R U {+oo) where X is a Eilberf space. We supply the dual S*with the weak topology. We suppose that the sequence I;, has an epi-limit 1'. Let us consider a subgradient p E aV(x)and let us choose sequences of elements xon and pon converging to x and p respectively and satisfying i) limsup,,, IX(xon) 5 V(X) (76) 1 ii) limsup,,, l.y(pon) 5 I,'*lp) (Such sequences do ezist by definition). Weintroduce the lack of consistency Then p belongs to the graphcal lower limit dbIiv:= (dIr)bof the subdiffer- ential dl', and we have the estimate Therefore, dV is the graphical limit of the subdifferentials dl<&. Proof We a,pply Ekeland's Theorem to the lower semicontinuous function I;; (y)- < PO,, y > for I a 86, >O € = 1 is any positive number if 6, 5 0 Since liminf,,, Vn*(pon)is finite, there exit a constant c and N large enough such that VnW(pon)5 c Vn2 N This implies that the functions y I- I;, (y) - < po,, y > are bounded below by -c thanks to the Fenchel inequality. Hence, we can a.pply Ekeland's Theorem: there exists a solution z,, satisfiing 15, (xtr)- < Pan 9 xn > +E 11x11 - XO,~II -< I:, (xot1) - < pot1 Son > (79 ri' ii) Vy€.YI 1k(x,,)- On the other hand, inequality (79)i)yields Taking into account that < p, x >= l,'(x)+ 'CT*(p)because p E d17(x)and that < pon, x, > 5 1', (x,,)+ lrn*(pon),we obtain If the right hand-side of this inequality is non positive, we infer that zo,, = z,, and. c being arbitrarely small, that d(pon,dl; (xO,,))= 0. If not, we deduce that llx - x,,11 5 c2/c = c. Hence the first part of the Theorem is proved. Since 1' is the epi-limit of the sequence I.:,, we deduce that each pair (r,p)in the graph of dlr is the limit of sequences (zon,pon)satisfying con- ditions (76).Hence it is also the limit of the sequences (z,, p, ) of the graph of aV, we have just constructed. This means that the subdifferential map dl' of the epi-limit of the func- tions V, is contained in the graphical lower limit of the subdifferential maps dlt',,. Since 81,' is a maximal monotone set-valued map when X is a Hilbert space, then the second part of the Theorem follows from Proposition ??. In particular, we deduce that under the assumptions of Theorem 10.1, the graphical convergence of the subdifferential al;;l to all' implies dl,' (r) = lim sup dl.; (x,) 11 -+a; However, we need some stability assumptions on the contingent second derivatives of the functions I;, for stating that some p E 81'(z) belongs to the ICuraton~skilower limit of dl:, (2, ). Naturally, the contingent second derivative C~~V(X,~)of IT at some point (r.p)in the graph of dV is defined as the contingent derivative of the set- valued map dlT at (x, p). Proposition 10.1 Let X be a Hilbert epace. Let us consider a sequence of eztended real-valued convez function8 I;', : X I- R U {+w ) and V whose domaine are not empty and sequences xo, and po, converging to r and p and eatiefying propert yier (76) Let p belong to dV(z). We poeit the following stability aeeumption: There eziet conatants e > 0, a E [0, 1[ and q > 0 euch that Then p E liminf dVn(zn) r1-CI) Remark The same proof yields the following result in the non convex case. Recall that the Clarke generalized gradient aV7(z)of 17 at z is defined by and dhat the subdifferential dOT;(r)is defined by Theorem 10.2 Let us consider a sequence of eztended real-valued lower eemicontinuoue function8 1.:' : ,X w R U {+w) and V whorse domainr are not empty. Let ue consider a eubgradient p E d"17(r).If there eziet RequenceR ro, and po, converging to r and p reepectively which eatiefy then p belong8 to the graphical lower limit abl/ := of the Clarke generalized gradient8 dl.', and we have the estimate (78)i 11 Asymptotic Paratingent and Circatangent Cones Let X be a topological vector space. We consider a sequence of subsets K,, c X of X and their Kuratowski upper and lower limits Kr and Kb. Definition 11.1 (Asymptotic Paratingent and Circatangent cones) If x E Kp we ehall eay that Kn - sn PL, (I) := limsup n+m,Ein3zn+3 hn ie the asymptotic paratingent cone to the Kuratowski upper Iimit Kd at s. We shall eay that the asymptotic circatangent cone to the Kura- toweka' lower limit Kbtx E Kb ie the set C;?,(x) defined by Kn - xn (83) cA-,(x) := n+cx,Kn~=n+tliminf hn Remark When we consider a constant sequence K, := K, we see that the asymptotic paratingent and asymptotic circatangent cones pi-, and Ck, coincide with the paratingent P$(z) and the Clarke tangent cone CK(x) respectively. It is easy to observe that the Kuratowski upper limit of the contingent cones to a sequence of Kn is contained in the asymptotic paratingent cone to the Kuratowski upper limit of the h','s: Proposition 11.1 Let us consider a eequence of subsets K, c ,TL' of X and their Kuratoweki upper Jimit Ks. Then lim sup Txn(z.) C pi-,(x) In +I The asymptotic circatangent cone t,o a Kuratowski lower limit is always a closed convex cone! and is actually equal to the Eiuratoa~skilower limit, of the contingent cones: Proposition 11.2 Let us conaidet a sequence of subsets K, c X of X and their Kuratowaki lower limit KP. 1. The arymptotic circatangent cone C;-,(r) is always a cloned conoez cone 2. If X is reflem've and is supplied with the weak topology, and if the aubeets h;, are weakly clo~ed,then , = liminf TKn(xn) n-raj,Kn3~n-i~ Proof 1. Let vl and v2 belong to Ckb(x). For proving that vl + v2 belongs to this cone, let us choose any sequence h, > 0 converging to 0 and any sequence of elements x,, E h', converging to x. There exists a sequence of elements el,, converging to trl such that the elements rl, := z, + h,vl, do belong to Kn for all n. But since rl, does also converge to x, there exists a sequence of elements v2, converging to v2 such that This implies that ol + vt belongs to Ckb(z)because the sequence of elements t~~,+ t:2, converges to vl + tr2. 2. Let us take x E Kband ty E liminf,,,,TKn(xn). We infer that, for all c > 0, z,,E h:, close to x and t small enough, inequalities v 2, E rEin(xn+ tl), 112 - 51) 5 2((x, + tt: - $11 imply that r, remains close to z, that there exists v, E Thin(2,) in a neighborhood oft?,so that for large enough n's. Let us set g,(t) := dh-, (xtl + tt,). Since g(.) is locally lipschitzean, it is almost everywhere differentiable. It implies'5 implies that g:,(r) 5 d(c,Th-,(xh-,(xn + 7125)) 5 E. We then integrate from 0 to t and get that, for all x,, E K, close to r and for all t €10, h] for some small enough positive h. We have proved that v belongs to c;, (x). 3. Let r belong to Ckb(r). Then, for all E > 0, there ezdst q > 0, N and B > O such that, for all h 5 8, n 2 N and x, E K, n B(x, q), dli,(z, + hr) 5 h,~ Then we can associate with such x,, elements y,h E Kt, such that We set t.,h := (y,h - x,)/h. Since 112.; - vll 5 2~ and since the space is reflexive, a subsequence (again denoted) c,h corrverges weakly to some tf, E v +~EB.Such a c, belongs to the contingent cone TKn(x,) (when X is supplied with the weak topology) and cornrerges to v. When the srtbsets K, are convex, we obtain the following telations: Proposition 11.3 Let us consider a sequence of convez subsets K, c S and its Kuratowski conaet upper and lower limits KP and Kb. Then: Proof 1. Let us take r E .Kb and v E Sp(x), the cone spanned KL x. Hence there exists h > 0 such that r + h,t- E hWP '"see I?, Proposition 4.1.3; Let us consider any sequence of elements x,, belonging to K,, and converging to r and let y, denote the projectmionof r + hv onto the closure of K,,. Since y, converges to r + ht., the sequence of elements t~, := (y,, - xn)/h converges to t.. Since belongs to K,, we infer that i:, belongs to the contingent cone Th-, (x,,). Hence 11, the limit of the tin's, belongs to the Kuratowslii lower limit. of the tangent cones TIin(rn) 2. Let r belong to Ii? and tq be chosen arbitrarily in the cone spanned bq' K.G - x. By denoting by x, and y, the projections onto h;, of x and x + h,c respectively, we infer that t3, := (y,, - xn)/h belongs to the tangent cone TKn(rn)and converges to o. Remark We deduce from Att,ouch's Theorem 10.1 that if the se- quence of convex subsets Kn has a limit K, then hrr; (2) = lim sup (x,,) n-m.zn-.t (We take for function 1: the indicators $K, of the K,, which epi-converges tso the indicator of K. M7e use then the fact that the subdifferential of an indicator is the normal cone.) When the dimension of X is finite, we deduce by transposition from Proposition 1.3 that if the sequence of convex subsets h', has a limit K, then limn Tn(,) = TK(x) 11-m.3,-3 We shall deduce from the Stability Theorem a formula for the asymp- totic circatangent cone of an inverse image. Theorem 11.1 Let .X and E' be two Banach spaces. We introduce a continuoue linear operator A E E(S,Y)and sequencee of cloeed subsets L, c S and AJ,, c E'. Let ue aseume that there etist constante c > 0, a E [O, 1[ and q > 0 such that Let x belong to the Kuratoweki lower limit of the sequence L,, n.4-'(,9/1,,) (which is equal to the intersection of the Kuratowski lower limit of L,, and the inverse image by A of the Kuratoweki lower limit of ,kit,). Then Proof Take any sequence of elements x, E L, n A-'lldn which converges to x. (We already know that x is the limit of such a sequence). Let us take any u E CLb(x)such that Au E CLfb(Ax).Hence for any sequence h,, > 0, there exist sequences u, and r;, converging to u and Au respectively such that, for all n 2 0, We apply now Theorem 3.2 to the subsets L,, x ,l~,,of A' x I' and the continuous linear operat:ors A Ej 1 associating to any (x,y) the element Ax - y, since we can write The stability assumptions of this Theorem are obviously satisfied. The pair (x,,+ h,u,, -42, + h,,v,,)belongs to L, x M, and (A6 1)(x, + h, u, , Ax, + h, t?, ) converges to 0 Ah Therefore, by Theorem 3.2, there exits a solution (x,, y,) E L, x Mnto the equation (AS 1)(z';;, y';;) = 0 such that Hence := (x, - jc^,)/h, converges to u, and we oobserve that for all n < 0, x, + h,u, belongs to L, n A-'(Af, j because x, + h,C = i% belongs to L, and .4(xn + h,u';;) = belongs to 114. We consider now the asymptotic paratingent cones to direct images. Theorem 11.2 Let S and I' be two Banach epacee. We introduce a con- tinuous linear operator A E L(S.E'), a sequence of closed subsets K,, c L c X and an element xo in the Kuratoweki upper limit h'u of the h;, '8. We assume that the restriction of A to L is proper from L to I-! so that .4(KF)= (.4(1i))E. We posit the following stability assumption: for all so E Kn.4-'(yo). there ezist constanta c > O! a E [O, 1[ and 7 > 0 such that Then, if S is reflezive and supplied with the weak topology, we have: (91) U AP;.~(x) = P;~,~,(AX) XEK~A-'(g) Proof ?Te always have the inclusion Conversely, let us take t7 E P;,(yo). Then there exist sequences of elements h, > 0. y, E A(Kn) and t?, E Y converging to 0, yo and t1 respectively such that We can write yn := Axo, where, A being proper, a subsequence (again denoted) xon converges do some XQ E K n A-'(yo). By Theorem 3.1, there exist a constant I' and solutions x, E K to the equation Ax, = y, + h.,vn such that Therefore. the sequence of elements u, is bounded, so that a subsequence (again denoted) u,, converges (weakly when the dimension of .ri' is infinite) to some ,u. Remark We can adapt the properness criterion given ky Theo- rem 5.1 for obtaining the following result: Proposition 11.4 Let S and Y be two Banach spacea. We introduce a continuous linear operator A E L(X.Y), a aequence of cloaed subseta K,, c S and an element xo in the Kuratowski lower limit ICD of the h', '3. We posit the following assumption: (95) o E Int I~(A*J+ u n I<; n A-;-~(X r, ) N>G,q>O n>N.~~fIinn(xo+rlB) 1 Then, if X is reflezicle and supplied with the weak topology, we have: U = Pi (,!, (As) xEGnA-I (g) Proof let US take u E Pi,(yo). Then there eGst sequences of elements h,, > 0: y, E A(K) and a, E Y converging to 0, yo and t. respectively such that, Let us consider solutions zo, E Kn to the equation Axo,, = y, and set u, := (5, - xO,,)/hn.We shall prwe that the sequences ro,, and u,, are point~4sebounded. Since the space S is reflexive, this -dl imply that subsequences (again denoted) xo,, and u,, converge to some xo E KnA-'(yo) and u respectively. so that u is an element of P:., (xo). For proving our claim, we associate with any p E X* an element q E E7*. an integer N such that, for all n 2 N. there eGst r,, E K: n ilriPn(x,,) satisfying p = A*q + r,,. Then < Pyxon >=< Q,Yn > + < rn,X~n> L IIQIII~Y~~~+ 1 because r, E Ki 1981 ( ii) < p, un >=< q. un > + < r., xO;ixn> 5 Ilqllllunll + 0 because r, E JN:-, (x,,) Therefore, our sequences XO,, and a,, are bounded. 12 Asymptotic Paratingent and Circatangent Epiderivatives We are now able to define asymptotic epi-derimtives of a sequence of func- tions I,,;, by taking the asymptotic tangent cones to their epigraphs. Definition 12.1 (Asymptotic Epiderivatives) Let u8 consider a eequence of eztended real-valued function8 'I< : S * R U {+oc) whose domaine are not empty and an element xo in the Kuratowski lower limit of the domains of the function8 1,;. We shall say that the function C."I'(xo) defined by (99) C~~*(XO)(U):= lim sup (\i(x+ h.ul) - A)/h n-+oo,r--*a,Vn(r) n + ca & (x, A, h) E Ep(Vn) x & converges to (XO,l;D(xo), 0) We deduce that the asymptotic circatangent epideridve is a positively homogeneous, lower semicontinuous and convex function form X to R U {-4u {+4. We shall estimate the asymptotic circatangent epiderivative of a family of functions Un := I< + Wn o A. Theorem 12.1 Let us consider two Banach space8 X and Y , a continuouu linear operator A E L(X,E'), and two sequences of eztended real-valued functions Vn and Wn form X and Y to R U {+oo) respectively. Let xo belong to the Kuratowski lower limit of the domains of the functions Un := vn + m; o A. We posit the following stability assumption: there ea'st constants c > 0: a.~[0,1[ and q > 0 such that, for all n, i) V z E Dom(1;) n B(xo,q). V y E Dom(VVn)n B(Axo,q) By c A (~om(~\(l'.)(2)) n CB~) -Dom(Dt (K) (y)) + ii) s"pv~Dom(D: lr.,,)(.)) / 5 C iii) s"pt~~D~m(~l(~7n)(r))t7) L c Then. the asymptotic circatange nt epiderivative of the sequence of functions U,,:= 1,; + I+', o A eatisfies the eetimate: Proof We apply Theorem 3.2 since we have seen tha.t, if we set i) K := Ep(lr)xEp(Fl.')xR. c .XxRxlr~RxR. ii) G(z,a, y, b, c) := (Az - y, a + b - c) . iii) H(x,a,y,b,c) := (x,c) we can write (103) Ep(U) = H(KnG-'(0,o)) The stability assumption of Theorem 3.2 being the same that the ones of Theorem 70. we already know that they can be derived from assumptions (21). Hence, we deduce that It remains to show that this inclusion implies inequalitj- (101). Let us set A = C)(l7)(s0)(u),p = C;(VI')(AX~)(AU)and v := A + p. Hence the element (u,A, Au, (I, v) belongs to By Theorem 70, it then belongs to the asymptotic circatangent cone to the subsets K, n G-'(0,O). Then, for all sequence h,, > 0, there esst elements (u,, A,, (I,) converging to (u,A, p) such that, for all n 2 N, Therefore, the pairs (x, + h,u,! a, + b, + h, (A, + (I,)) belong to the epigraph of U,. Since (u,. A, + (I,) converges to (u. v), we deduce that References [I] ARTSTEIK Z. 8L WETS R. ( 1986) Approzimating the integral of a multifunction (21 ATTOIJCH H. 8L WETS R. (to appear) Ieometriee for the Legendre-Fenchel tranlrform Trans. A.hi.S. [3] ATTOIJCH H. 8: WETS R. (1981) Approzimation and eonver- ge nce in nonlinear optimization Nonlinear Programming, 4, 367-394 [4] ATTOIJCH H. 8: WETS R. (1981) Approzimation and Conver- gence in Nonlinear Optimization In NONLINEAR PROGRAM- MING 4, eds, Mangaserian O., Meyer R, P Robinson S., 367-394 151 ATTOUCH H. & WETS R. (1983) A convergence for bi- aaraite function8 aimed at the convergence of #addle valuep In MAHTEMATICALTHEORIES OF OPTIMIZATIONeds. 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(1987) Smooth and heavy solutions to control prob- lems Proceedings of the conference on Functional Analysis at Santa Barbara, June 24-26, 1985, (IIASA U'P-85-44) (141 AlTBIK J.-P. (to appear) VIABILITYTHEORY [15] BEER G. (1987) Metric epacee with nice cloeed balle and dis- tance functione for cloeed set8 Bull Austral. Math. Soc., 35, 81-96 [16] BREWS H. (1973) OPERATELRSMAXIMAU); MONOTONES ET SEMI-GROUPES DE CONTRACTION DANS LES ESPACES DE HILBERT North-Holland (171 CASTAING C. & VALADIER M. (1977) CONVEX ANALYSIS AND MEASURABLE MULTIFUNCTIONS Springer-Velag, Lecture Notes in Math. # 580 (181 CHOQUET G. (1947-1948) Convergencecr Annales de 1' Univ. de Grenoble, 23, 55-112 1191 CLAME F. H. (1983) OPTIM~ZATIO~VAND NONSMOOTH ANALYSIS V7iley-Interscience 1201 de GIORGI E. 8t FRAKZOh7 T (1975) Su un tipo de conver- genza van'azionale Atti Acc. Naz. Lincei. 58. 842-850 (211 de GIORGI E. (1975) Sulla convergenza di alcune eucceeeioni di integrali del tipo del area Rend. Mat. ITniv. 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(1979) Convergence od dequenees of cloeed set8 Topology Proceed., 4, 149-158 [37] SALINETTI G. & WETS R. (1979) On the convergence of se- quences of conaez sets in finite dimensions SIAM review, 21. 18-33 [38] SALINETTI G. & WETS R. (1981) (3n the convergence of cloeed-valued meaeurable multifunctions Trans. of AM'S. 266, 275-289 [39] SXLINETTI G. & WETS R. (1987) Weak convergence of prob- ability meaeune revieited IIASX WP 87-30 [40] SHI SHUZHONG (1987) Choquet Theorem and Nonemooth Analyeie Cahiers de I~Iathdmatiquesde la Dhcision [41] SHI SHUZHONG (1987) Thkorkme de Choquet et Analyse non rkgulikre Comptes-rendus de 1'Acadimie des Sciences, PARIS, 305, 41-44 [42] WETS R. (1982) A formula for the level set8 of epi-limits and some applicatione In >IAHTEMATIC.%LTHEORIES OF 0 PTI- SIIZATION eds. Cecconi P. & Zolezzi, Springer-Verlag, Lecture Notes in Mathematics #979. 256-268 1431 WETS R. (1982) Convergence of sequences of closed functionv In PROCEED.SYMPOSIU~~ OON%I.IHTEXIATIC:AL PROGRA~~- MING AND DATA PERTURBATIONS,eds. Fiacco .i.,16-27 [44] WETS R. (1984) On a compactness theorem for epiconvergent sequences of functions In ~I.ITHEMATICALPROC;RAMX[ING. eds COTTLE, LE&IXNSOX U' KORTE, 'North-Holland, 347- 35 1 [45] WETS R. (1986) An elementary proof of Minkoweki'e theorem for 3ystem of linear inequalitiee Index Asymptotic Epideri\'itti\?es 30 -4.q-mptoticEpiderivatives 50 Asymptotic Paratingent and Cir- catangent cones 44 Attouch's Theorem 39 Convergence Theorem 26 Epi-limits 28 Graphical Convergence 20 Graphical upper and lower lim- its of set,-valued maps 20 Inverse Function Theorem 14 Inverse Stability Theorem 12 Kuratoa~ski'limits 5 Iiuratom?ski's upper and lower limits of set-valued maps 5 Lim sup inf 29 Limits of Infima 33 Pointwise Convergence 20 Stability of solution maps 25 Stability of viability domain 25 Variational system 32